Chapter 5. Root Locus Techniques
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1 Chapter 5 Rt Lcu Techique
2 Itrducti Sytem perfrmace ad tability dt determied dby cled-lp l ple Typical cled-lp feedback ctrl ytem G Ope-lp TF KG H Zer -, - Ple 0, -, - K Lcati f ple eaily fud Variati f fgai K d t affect the lcati f ay ple H Cled-lp TF KG T KG H K 6 K 8 K K Lcati f ple eed t factr the demiatr difficult variati f gai K d chage the lcati f ple Eay way t kw the CL ple lcati w/ lvig higher-rder characteritic equati f CLTF fr variu gai K? Rt lcu graphical repreetati f the cled-lp ple a a fucti f ytem parameter Ca be ued t deig ytem parameter f the high rder ytem t yield a deired ytem pecificati Etimatig cled-lp ple lcati whe gai K i varied uig pe-lp ple Repreet the ple f T a K varie
3 Vectr Repreetati f Cmplex Number Vectr repreetati f cmplex umber Cmplex umber σ jω Magitude M ad agle, a M Cmplex fucti F a whe σ j ω Cmplex fucti F ha a zer at -a F a σ a jω a cmplex umber a ca be repreeted by a vectr draw frm the zer f the fucti t the pit Ex F 7 5 j
4 Cmplicated fucti m zi i : prduct F where M: umber f zer pi N: umber f ple j Value f F at ay pit Each cmplex factr vectr magitude & agle Magitude f F M where zer ple legth legth z i p i m zi i p j i Magitude f the vectr draw frm t the pit p Magitude f the vectr draw frm t the pit z i p i Agle f F at ay pit Meaured frm the pitive real axi agle t zer m i z i j p agle t i ple
5 Ex Fid F at the pit - j F Magitude M zer legth ple legth j j j Agle r F agle t zer z p 0 p agle t ple ta F ta ta ta.
6 Prpertie f Rt Lcu Subject trackig camera ytem Kb 에해당 Equivalet cled lp TF Cled lp ple fucti f gai K P, 0 ± 0 K
7 Lcati f C.L.ple vary with gai K Sytem perfrmace varie K ple K<5 Real : verdamped Ple plt K5 Repeated real: critically damped K>5 uderdamped d d Cectig the C.L.ple crrepdig t frm K 0 t K C.L. ple lcu fr frm K 0 t K Rt lcu Rt lcu Path f cled-lp ple a gai i varied Uig rt lcu, we ca eaily aalyze the characteritic f the higher-rder ytem w/ calculatig cled-lp ple
8 Prpertie f Rt Lcu Higher-rder ytem difficult t calculate the cled-lp ple lcati fr variu gai K. Uig rt lcu, withut lvig demiatr plymial f cled lp TF, it i pible t have rapid ketch f cled-lp ple lcati chage fr variu gai K rt lcu C.L.T.F. KG T KG H KG KL where L G H :OLTF Characteritic equati i.e. C.L. ple the value that atify the characteritic equati KG H 0 r KG H k 80 KG H k 80 KL 0 k 0, ±, ±, ±, L Ay cmplex umber that atifie thee cditi cled-lp ple! KG H K G H Agle cditi determie the rt lcu Magitude cditi Determie the pecific piti crrepdig t pecific value f K the rt lcu
9 Ex Ope-lp TF KG H K Cled-lp TF T K K 7K K If a ple a jb i a cled-lp ple fr me value f gai K a jb mut atify KG H k 80 ad KG H Cider a the firt tet pit KG H j agle t zer 56. agle t i t a cled-lp ple j ple j What abut / j KG H 80 j K G H / i a cled-lp ple fr me value f gai K ple legth zer legth j / j j / j / / j / i a cled-lp ple i.e. i a pit the rt lcu whe K 0.
10 Ple ad Zer at Ifiity Ifiite ple: if O.L.T.F. apprache, a apprache OLTF ha a ple at ifiity Ifiite zer: if O.L.T.F. apprache 0, a apprache OLTF ha a zer at ifiity ex G G ha a ple at ifiity ice ha a zer at ifiity limg lim lim G lim 0 Ex KG H K Three fiite ple 0, -, - N fiite i zer K lim KG H 0 Fr every fucti f umber f fiite i ple ifiite i ple umber f fiite zer ifiite zer Three zer at ifiite fiite ple 0 ifiite ple 0 fiite zer ifiite zer
11 Shw hw lcati f the CL ple mve a K varie frm 0 t Sketchig the Rt Lcu K0~ 변할때 rt lcu가어떻게변하는지본다.. Mark pe lp ple with x ad pe lp zer with. Draw rt lci the real-axi t the left f a dd umber f real ple plu zer K>0일때, 실수축상에서 Tet pit의오른쪽에있는 ple과 zer의수의합이홀수이면그점은rt luc상에있다 Satifyig the agle cditi KG H k 80. Rt lcu i.e. K 0 tart at fiite ad ifiite pe-lp ple ad ed at fiite ad ifiite pe-lp zer i.e. K. Draw the aymptte f rt lcu a. Real-axi itercept r ceter f aymptte fiite ple fiite zer σ a # fiite ple# fiite zer b. 양의실수축과이루는각도 k 80 a k 0, ±, ±, ± # fiite ple# fiite zer Nte. - Number f brache f the rt lcu umber f cled-lp lp ple ytem rder Brach: path that e ple travere - The rt lcu i ymmetrical abut the real axi
12 ex 8. Ed at zer at tifiity it. Mark OL ple ad zer. Draw lci real axi t the left f dd umber. Start at ple ad ed at zer. Draw aymptte σ a a 0 # π / π fiite k 80 ple# fiite fr fr k 0 k 5 π / fr k π / zer Ed at zer at ifiity 5π / π Check! # f aymptte # fiite ple - #fiite zer # f brache # f CL ple ytem rder The rt lcu i ymmetrical abut the real axi # ifiite zer #fiite zer #fiite ple Ed at zer at ifiity
13 Refiig the Sketch breakaway frm real axi ad mve it the cmplex plae A. Real-axi Breakaway ad Break-i Pit σ σ : Breakaway pit : Break-i Pit 80 Break-i ad Breakaway agle : : # f CL ple arrivig at r departig frm the BA r BI the real axi A K icreae Tw ple 90 at BA Tw zer 90 at BI Determie the lcati f BA ad BI pit O the real axi betwee pe-lp p ple, gai K i maximum at BA pit O the real axi betwee pe-lp zer, gai K i miimum at BI pit dk d σ 0 where σ i a either BA r BI pit K Q KG H 0 G H Characteritic equati
14 Ex K K H KG 5 8 K dk σ σ σ σ d dk σ σ σ σ A d i.8.5, σ BA ad BI pit
15 B. j ω -Axi Crig A pit the rt lcu that eparate the table perati f the ytem frm the utable perati A ple j ω -axi at a certai gai K Hw t fid the value f ω ad K? Methd I Subtitute j ω directly it characteritic equati 0 A ple j ω -axi σ jω Methd II Ue Ruth table A rw f etire zer f Ruth table ple j ω-axi pible
16 Ex fid the frequecy ad gai K fr which the rt lcu cre the imagiary axi. C.L.T.F. T 7 K 8 K K Methd I Characteritic equati: 7 8 K K 0 A j ω i a ple f cled-lp ytem, it mut atify characteritic equati Subtitute t j ω it characteritic ti equati jω 7 jω jω 8 K jω K 0 ω j7ω ω 8 K jω K 0 Real part Imagiary part ω ω K 0 7ω 8 K ω 0 Slve fr ω ad K ω ±.59 K 9.65 Rt lcu cre j ω -axi at ± j. 59 at a gai f K 9.65
17 Methd II Ue Ruth table K T 7 8 K K N ig chage frm t LHP ple remaiig, but eve plymial f require ymmetric ple ple huld be j jωω -axi Oly rw ca make all zer rw K 65K 70 0 Aumig K>0, the K 9.65 Frm eve plymial uig rw 90 K K ± j K 9.65 j.59 Rt lcu cre j ω-axi at ± j. 59 at a gai f K 9.65 K 0 K 0 Sytem i table fr 0 K < 9.65 K 9.65 j.59
18 C. Agle f Departure ad Arrival Rt lcu tart at pe-lp ple ex. p ad ed at fiite ad ifiite pe-lp zer ex. z Fr mre accurate rt lcu, eed t kw the rt lcu departure agle frm the cmplex ple arrival agle t the cmplex zer If a cmplex umber i rt-lcu ad cle ε t a cmplex ple p, i.e. i e f the cledlp ple, it mut atify agle cditi Tet pit KG H agle t zer k 80 6 agle t 5 ple if wat t kw the departure agle frm a cmplex ple p 6 5 k 80 Nte. if ple p ha multiplicity q, the departure agle KG H k 80 q k q 5 80
19 if wat t kw the arrival agle t a cmplex zer z z Agle cditi fr a cmplex umber ear z ple agle t zer agle t H KG 80 k k k N t if h lti li it th d t l Nte. if zer z ha multiplicity q, the departure agle q k H KG q k Nte. ad 대신각을찾고자하는 ple 이나 zer 까지의각도를직접구하면된다.
20 Ex fid the agle f departure frm the cmplex ple ad ketch the rt lcu Ope lp ple at -, -±j Departure agle frm -j k 80 k 80 ta ta 80
21 D. Plttig ad Calibratig the Rt Lcu Hw t fid exact pit at which rt lcu atifie certai cditi ad the gai at that pit? Methd I Ex Fid exact pit at which rt lcu cre the 0.5 dampig lie ad the gai at that pit K KG H ζ c c O the ζ 0.5 lie, ly a pit with radiu r 0.77 atifie agle cditi KG H k 80 5 zi i G H p m j i zer legth G H ple legth ple legth A C D E At that pit, the crrepdig gai K. 7 G H zer legth B
22 Methd II Ex Fid exact pit at which rt lcu cre the 0.58 dampig lie ad the gai at that pit G 6 C R G G α 0 ζω ω 0.76ω α α 0.76ω ω 0.76ω α ω αω c 0.58 ζ ω α ω α ω αω ω.8 ζ 0.58 d.007 ± j.67 CL ple pair ζ 0.58 dampig lie C B A ζω ± jω ζ.007 ± j K ple legth G H zer legth A B C 80
23 Cmmet T be table All C.L. ple mut be i LHP If #O.L. ple - #fiite zer There i a vlue f the gai K beyd which rt rci eter the RHPi..e ytem becme utable
24 Nmiimum-Phae Sytem Skip! Miimum phae ytem: all the ple ad zer lie i LHP Nmiimum phae ytem: at leat e ple r zer lie i RHP G K T a T H & T > 0 a G K T a T T 80 T ±80 k Phae hift thi i why it i called miimum phae K T ± 80 T K a a k 0 Lcu Nt here!
25 Geeralized Rt Lcu S far, rt lcu a a fucti f the frward-path gai K Hw t draw rt lcu fr variati f ther parameter? Ex rt lcu fr variati f the value f pe lp ple p? 0 KG H gai K wa a multiplyig factr f the fucti but P i t. p Demiatr f C.L.T.F. fr a rt lcu fr gai K variati KG H Fr a rt lcu fr p variati, we eed a CLTF demiatr pg H KG T KG H KG p 0 p 0 0 Ilatig p 0 p 0 0 p 0 p H 0 P i w a multiplyig factr f the pe lp trafer fucti G H 0 Path f cled lp ple a p i icreaed
26 Summary f the Step fr Ctructig Rt Lci. Cvert it a frm i which ytem parameter i a multiplyig factr. Lcate the pe-lp ple ad zer the -plae. The brache tart frm pe-lp pple ad ed at fiite r ifiite zer. Draw the rt lcu the real axi 5. Determie the aymptte f rt lcu 6. Fid the breakaway ad break-i pit 7. Dt Determie the agle f fdeparture/arrival lf frm/t a cmplex ple/zer l/ 8. Fid the pit where the rt lci may cr the imagiary axi 9. Lcate the cled-lp ple the rt lcu ad determie the crrepdig gai K by ue f the magitude cditi
27 Rt Lcu fr Pitive-Feedback Sytem KG T KG H Pitive feedback ytem A cled-lp ple exit whe KG H 0 KG H 60 k k 0, ±, ±, ± Agle cditi fr a rt lcu f pitive feedback ytem KG H 60 k Rule fr ketchig rt lcu Read by yurelf
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